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Chapter 3: Problem 59

Concept Check In Exercises \(55-62,\) describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ 3 y=-6 $$

Short Answer

Expert verified
The graph is a horizontal line at y = -2.

Step by step solution

01

- Simplify the Equation

Divide both sides of the equation by 3 to isolate y. This transforms the equation into: \[ y = \frac{-6}{3} \] Simplify the fraction: \[ y = -2 \]
02

- Recognize the Type of Line

The simplified equation \( y = -2 \) represents a horizontal line where the value of y is constant.
03

- Describe the Graph

This equation describes a horizontal line that passes through the point \( (0, -2) \) and is parallel to the x-axis. This means that at every point along the line, the y-value will be -2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

simplifying linear equations
When simplifying linear equations, the goal is to isolate one variable to make the equation easier to understand.
Take the example from the problem: \[ 3y = -6 \]
To isolate y, divide both sides by 3. This step simplifies the equation: \[ y = \frac{-6}{3} \]
After simplifying, we get: \[ y = -2 \]
Now, the equation is simpler and we can easily interpret it. Simplification helps us to understand the basic structure of the equation and makes it easier to graph or solve.
horizontal lines
A horizontal line in the coordinate plane is a straight line that runs from left to right, parallel to the x-axis.
In our example, the equation simplifies to: \[ y = -2 \]
This is a specific type of linear equation where y is always equal to a constant value. For horizontal lines:
  • The slope is 0
  • The line crosses the y-axis at \(y=-2\)

Therefore, no matter what the x-value is, the y-value remains constant at -2. This results in a flat, unchanging line.
coordinate plane
The coordinate plane is a two-dimensional surface used to graph equations.
It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
  • Points are described by coordinates (x, y)
  • For instance, the point (0, -2) lies on the y-axis, 2 units below the origin

In the given problem, after simplifying the equation to \[ y = -2 \]
The graph will be a horizontal line that crosses the y-axis at -2. This line remains parallel to the x-axis and represents all points where the y-coordinate is -2.

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Most popular questions from this chapter

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